The present invention relates to a new and distinct raspberry variety designated ‘DrisRaspThree’ and botanically known as Rubus idaeus L. This new raspberry variety was discovered in Santa Cruz, Calif. in October 2003 and originated from a cross between the proprietary female parent raspberry plant ‘X146.7’ (unpatented) and the proprietary male parent raspberry plant ‘S858.1’ (unpatented). The original seedling of the new variety was first asexually propagated by tissue culture at a nursery in Santa Cruz, Calif. ‘DrisRaspThree’ was subsequently asexually propagated by root cuttings and underwent further testing at a nursery in Santa Cruz, Calif. for four years. The present invention has been found to be stable and reproduce true to type through successive asexual propagations.